Mathematical Sciences: Dynamics and Patterns in Reaction Diffusion Equations

点击翻译按钮获取中文摘要

Reaction diffusion equations may give rise to a variety of pattern like solutions. These include traveling wave solutions, stable steady solutions, cellular patterns, oscillatory solutions, and even solutions which behave in a chaotic manner. All of the systems which the principle investigator will consider arise as a mathematical model of a biological or chemical system. This project will consider models for combusion, chemical reactions on catalytic surfaces, phase transitions, and bursting in excitable membranes. The primary goals of this research project are to investigate through which mathematical mechanisms these patterns may arise, analyse the stability properties of each pattern, and understand what physical significance each pattern has for the underlying physical problem. The problems to be addressed in this project raise a number of fundamental mathematical issues. Of particular interest are patterns arising in more than one space dimension. Much of the analysis will be concerned with how higher dimensional patterns bifurcate from basically one dimensional objects such as planar wave solutions and radially symmetric solutions. Simple examples demonstrate that the structure of the bifurcating patterns may be incredibly complex.